The present invention relates to permutation groups and games utilizing such groups.
If one defines a set of elements X and an operation * that assigns to each pair of elements a and b of X an element c of X, then the pair G=(X,*) is called a group if it has the properties of closure, associativity, identity and inverse. For the pair (X,*) to have closure, the operation * must assign to each pair of elements of X another element of X. Thus, if a, b are elements of X, then *(a,b) (which may also be written a*b) must also be an element of X. For the pair (X,*) to have associativity, then a*(b*c)=(a*b)*c where a, b, c are elements of X. For the pair (X,*) to have an identity, there must be an element I in X such that I*x=x*I=x for each element x of the set X. For the pair (X,*) to have an inverse, then each element x in the set X must have an element x.sup.-1 in the set X for which x*x.sup.-1 =x.sup.-1 *x=I where I is the identity element.
An example of a group is the pair formed by the positive real numbers and the operation of multiplication. Multiplication of two positive real numbers has closure since it always yields a positive real number. Multiplication is associative; the identity element is 1; and the inverse of any positive real number "a" under the operation of multiplication is 1/a. Another example of a group is the pair formed by the real numbers and the operation of addition.
A permutation of a set of elements is an ordering of the set of elements. For example, if the set of elements consists of the four members, 1,2,3 and 4, one such ordering is 1234 and another such ordering is 2143. The number of different orderings of a set of elements is equal to n!, where n is the number of different elements in the set. For example, if the set of elements consists of the four numbers 1,2,3,4, then there are 4!=4.times.3.times.2.times.1 =24 different ways of arranging these numbers. These 24 different ways are set forth in Table I.
TABLE I ______________________________________ 1234 2134 3124 4123 1243 2143 3142 4132 1324 2314 3214 4213 1342 2341 3241 4231 1423 2413 3412 4312 1432 2431 3421 4321 ______________________________________
While there are 24 different ways of arranging four numbers, if we regard each set of numbers as "wrapping around" such that the fourth number is next to the first number then the 24 different patterns each of which has four different starting points Thus we may say that there are only six rotationally distinct ways of arranging the four numbers. For example, for the set "1,2,3,4" of column 1, there appears a rotational equivalent in columns 2,3 and 4, namely "2,3,4,1", "3,4,1,2", and "4,1,2,3". Similarly, for the set "1,2,4,3" in column 1, there appears a rotational equivalent in columns 2,3, and 4, namely "2,4,3,1", "3,1,2,4", and "4,3,1,2", and so on.
As is demonstrated below, operations can be defined on the collection of all permutations of a set of elements such that the pair formed by the collection and the operation(s) satisfies the properties of closure, associativity, identity and inverse. Such pairs are called permutation groups. For further information about permutation groups, see Fred S. Roberts, Applied Combinatories, (Prentice-Hall, 1984), especially .sctn.7.2.
In the teaching of the rules of permutation groups to beginning students and others having trouble mastering the concepts and principles of same, it is important for teachers to present the material in an effective manner. Traditional methods of teaching such as memorization of modular systems and derivation of equations has in many instances been very difficult for both the student and the teacher. It is therefore desirable to have an apparatus and a method for teaching and learning the rules of permutation groups which is less tedious than the traditional methods and which provides for the student a rewarding experience.
Equally important are the avid game players who are always looking for new and challenging games which may be played for sheer intellectual stimulation and pleasure. It is therefore desirable to have an apparatus and a method for playing a game which has varying degrees of difficulty and which provides exciting entertainment to the avid game player.